A240711 Sum of the largest parts in the partitions of 4n into 4 parts with smallest part = 1.
1, 15, 62, 163, 333, 596, 973, 1475, 2130, 2959, 3969, 5192, 6649, 8343, 10310, 12571, 15125, 18012, 21253, 24843, 28826, 33223, 38025, 43280, 49009, 55199, 61902, 69139, 76893, 85220, 94141, 103635, 113762, 124543, 135953, 148056, 160873, 174375, 188630
Offset: 1
Examples
For a(n) add the parts in the first columns.
13 + 1 + 1 + 1
12 + 2 + 1 + 1
11 + 3 + 1 + 1
10 + 4 + 1 + 1
9 + 5 + 1 + 1
8 + 6 + 1 + 1
7 + 7 + 1 + 1
11 + 2 + 2 + 1
10 + 3 + 2 + 1
9 + 1 + 1 + 1 9 + 4 + 2 + 1
8 + 2 + 1 + 1 8 + 5 + 2 + 1
7 + 3 + 1 + 1 7 + 6 + 2 + 1
6 + 4 + 1 + 1 9 + 3 + 3 + 1
5 + 5 + 1 + 1 8 + 4 + 3 + 1
7 + 2 + 2 + 1 7 + 5 + 3 + 1
5 + 1 + 1 + 1 6 + 3 + 2 + 1 6 + 6 + 3 + 1
4 + 2 + 1 + 1 5 + 4 + 2 + 1 7 + 4 + 4 + 1
3 + 3 + 1 + 1 5 + 3 + 3 + 1 6 + 5 + 4 + 1
1 + 1 + 1 + 1 3 + 2 + 2 + 1 4 + 4 + 3 + 1 5 + 5 + 5 + 1
4(1) 4(2) 4(3) 4(4) .. 4n
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1 15 62 163 .. a(n)
Links
- Index entries for sequences related to partitions
- Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).
Programs
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Mathematica
b[n_] := Sum[((4 n - 2 - i)*Floor[(4 n - 2 - i)/2] - i (4 n - 2 - i)) (Floor[(Sign[(Floor[(4 n - 2 - i)/2] - i)] + 2)/2]), {i, 0, 2 n}]; c[1] = 1; c[n_] := Sum[Sum[i (Floor[(Sign[(Floor[(4 n - 2 - j)/2] - j)] + 2)/2]), {i, j + 1, Floor[(4 n - 2 - j)/2]}], {j, 0, 2 n}]; Table[b[n] - c[n], {n, 50}] -
PARI
Vec(x*(7*x^6+27*x^5+43*x^4+52*x^3+33*x^2+13*x+1)/((x-1)^4*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Apr 11 2014
Formula
G.f.: x*(7*x^6+27*x^5+43*x^4+52*x^3+33*x^2+13*x+1) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, Apr 11 2014